Inclusion Criteria:

          -  Patients must provide informed written consent

          -  Eastern Cooperative Oncology Group (ECOG) performance status 0-1

          -  Clinical stage IV invasive mammary carcinoma

          -  For phase 1b: HER2 negative, as defined for phase II; any ER/PR (negative or positive)
             can be enrolled in the phase 1b portion

          -  For phase II: ER negative (defined as expression of ER in =< 1% cells), PR negative
             (defined as expression of PR in =< 1% cells), HER2 negative (acceptable methods of
             HER2 analysis include IHC [0, 1+], fluorescence in situ hybridization [FISH] with
             HER2/centromere on chromosome 17 [CEN17] ratio < 2, and/or chromogenic in situ
             hybridization [CISH] with HER2/CEN-17 ratio < 2), as previously documented by
             histological analysis

          -  Androgen receptor positivity, defined as >= 10% of tumor cell nuclei with
             immunoreactivity for AR on central review at Vanderbilt

          -  Measurable or bone-only evaluable disease; measurable disease is defined as at least
             one lesion that can be accurately measured in at least one dimension by Response
             Evaluation Criteria in Solid Tumors (RECIST) criteria 1.1, with radiologic scans
             within 21 days of day 1, cycle 1

          -  Any number of prior therapies as long as patients have adequate performance status and
             meet all other eligibility criteria

          -  Prior treatment with anti-androgens other than enzalutamide is acceptable

          -  Phase 1b only: Formalin-fixed paraffin embedded blocks (FFPB) or fresh frozen tissue
             from the original diagnosis or the metastatic setting should be located; tissue must
             be submitted with 3 weeks of study initiation

          -  Phase II only: Biopsy of a metastatic lesion in patients with reasonably accessible
             metastatic lesions (chest wall, skin, subcutaneous tissue, lymph nodes, skin, breast,
             bones, lung, and liver metastases); if a reasonably accessible metastatic lesion is
             not available, the patient may go on study provided that archived tissue is available;
             however, if a reasonably accessible site is available for biopsy, the patient must
             agree to biopsy; any patients not undergoing biopsy must be approved for study
             enrollment by the Protocol Chair; biopsies may be done with local anesthesia or
             intravenous conscious sedation, according to institutional guidelines; if a biopsy
             requires general anesthesia, then it is only allowed if acquisition of tissue is
             clinically indicated, and excess tissue may be collected for research purposes;
             patients without sites available for biopsy must have available tissue (archived
             formalin-fixed paraffin embedded blocks [FFPB] or fresh frozen tissue from original
             diagnosis or metastatic setting) for correlative studies; tissue needs to be located
             and available at the time of registration (tissue needs to be submitted within 3 weeks
             of study initiation)

          -  Patients must have adequate hematologic, hepatic, and renal function. All tests must
             be obtained within 28 days of starting treatment. Labs are to be repeated on cycle 1,
             day 1 and must still meet eligibility. These include:

          -  Absolute neutrophil count (ANC) >= 1500/mm^3

          -  Platelet count >= 75,000/mm^3

          -  Hemoglobin (HgB) >= 9 g/dL

          -  Creatinine =< 1.5 X upper limits of normal (ULN)

          -  international normalized ratio (INR) ≤2

          -  Total serum bilirubin =< 1.5 x ULN (in patients with known Gilbert syndrome, a total
             bilirubin =< 3.0 x ULN, with direct bilirubin =< 1.5 x ULN)

          -  Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =< 3 x ULN (or =<
             5.0 x ULN if hepatic metastases are present)

          -  For patients without known type II diabetes, the following is required at screening:

               -  Fasting plasma glucose =< 160 mg/dL (7.49 mmol/L) and glycosylated hemoglobin
                  (HbA1c) < 7.5 % or International Federation of Clinical Chemistry (IFCC) < 53
                  mmol/mol

          -  For patients with type II diabetes receiving only oral anti-hyperglycemic therapy
             (patients receiving insulin are not eligible), the following are required at
             screening:

               -  HbA1c < 8.5 % or IFCC < 69.4 mmol/mol

               -  Stable regimen of oral anti-hyperglycemic therapy without insulin usage for at
                  least 3 weeks prior to first study treatment

               -  Fasting plasma glucose levels =< 160 mg/dL (8.88 mmol/L) and no hypoglycemia
                  (blood sugar [BS] < 60) during home monitoring for at least 1 week prior to study
                  entry

          -  Patients must be able to swallow and retain oral medication

          -  For patients who are not postmenopausal or surgically sterile (absence of ovaries
             and/or uterus), agreement to remain abstinent or to use two adequate methods of
             contraception (e.g., condoms, diaphragm, vasectomy/vasectomized partner, tubal
             ligation), during the treatment period and for at least 30 days after the last dose of
             study treatment or 3 months after discontinuation of taselisib and/or enzalutamide,
             whichever is longer; hormone based oral contraceptives are not allowed on study;
             postmenopausal is defined as:

               -  Age >= 60 years

               -  Age =< 60 years and amenorrheic for 12 months in the absence of chemotherapy,
                  tamoxifen, toremifene, or ovarian suppression; or follicle stimulating hormone
                  and estradiol in the postmenopausal range

          -  Patients may have received radiation therapy to painful bone metastases or areas of
             impending bone fracture as long as radiation therapy is completed >= 2 weeks prior to
             day 1 of cycle 1 of treatment; patients who have received prior radiotherapy must have
             recovered from toxicity (=< grade 1) induced by this treatment; baseline radiologic
             scans must be obtained after completion of radiation

          -  Patients must complete all screening assessments

        Exclusion Criteria:

          -  Any kind of malabsorption syndrome significantly affecting gastrointestinal function,
             including history of Crohn's disease or inflammatory bowel disease

          -  Concurrent anti-cancer therapy (chemotherapy, radiation therapy, surgery,
             immunotherapy, hormonal therapy, biologic therapy) other than the ones specified in
             the protocol; patients must have discontinued the above cancer therapies for 1 week
             prior to the first dose of study medication, as well as recovered to baseline from
             toxicity induced by previous treatments; any investigational drugs should be
             discontinued 2 weeks prior to the first dose of study medication and radiotherapy must
             have been completed >= 2 weeks prior to initiation of study drug (cycle 1, day 1)

          -  Prior use of PI3K or Akt inhibitors in the metastatic setting for the treatment of
             cancer; these include, but are not limited to: taselisib, GDC-0941, GDC-0980, BEZ235,
             BKM120, LY294002, PIK-75, TGX-221, XL147, XL765, SF1126, PX-866, D-87503, D-106669,
             GSK615, CAL101; patients who have received PI3K/Akt inhibitors previously for < 4
             weeks will be eligible

          -  Prior treatment with enzalutamide

          -  Current or previously treated brain metastasis or active leptomeningeal disease; head
             imaging is required during screening in all patients to exclude the presence of
             central nervous system (CNS) metastatic disease

          -  History of seizure or any condition that may predispose to seizure; history of loss of
             consciousness or transient ischemic attack within 12 months before day 1

          -  Pregnant or lactating women

          -  Insulin-dependent diabetes; patients with type II diabetes must meet the inclusion
             criteria outlined above

          -  Uncontrolled intercurrent illness including, but not limited to:

               -  Ongoing or active infection requiring parenteral antibiotics

               -  Impairment of lung function (chronic obstructive pulmonary disease [COPD] > grade
                  2, lung conditions requiring oxygen therapy) or current dyspnea at rest

               -  Symptomatic congestive heart failure (class III or IV of the New York Heart
                  Association classification for heart disease)

               -  Known left ventricular ejection fraction (LVEF) < 50%

               -  Unstable angina pectoris, angioplasty, stenting, or myocardial infarction within
                  6 months

               -  Uncontrolled hypertension (systolic blood pressure > 160 mm Hg or diastolic blood
                  pressure > 100 mm Hg, found on two consecutive measurements separated by a 1 or
                  2-week period despite adequate medical support)

               -  Clinically significant cardiac arrhythmia (multifocal premature ventricular
                  contractions, bigeminy, trigeminy, ventricular tachycardia that is symptomatic or
                  requires treatment [National Cancer Institute-Common Terminology Criteria for
                  Adverse Events, version 4.0, grade 3])

               -  Corrected QT using the Fridericia correction formula (QTcF) >= 480 msec on
                  screening electrocardiogram (EKG)

               -  Known history of QT/correct QT (QTc) prolongation or Torsades de Pointes (TdP)

               -  ST depression or elevation of >= 1.5 mm in 2 or more leads

               -  Diarrhea of any cause >= Common Terminology Criteria for Adverse Events (CTCAE)
                  grade 2

               -  Active autoimmune disease that is not controlled by nonsteroidal or steroidal (<
                  10 mg of prednisone per day) anti-inflammatory drugs or active inflammatory
                  disease, including small or large intestine inflammation such as active Crohn's
                  disease or ulcerative colitis, which requires immunosuppressive therapy

               -  Psychiatric illness/social situations that would compromise patient safety or
                  limit compliance with study requirements including maintenance of a
                  compliance/pill diary

               -  Known history of chronic liver disease including cirrhosis, current alcohol
                  abuse, or infection with hepatitis B virus or hepatitis C virus (active or
                  carrier) or renal failure

               -  Known history of chronic pancreatitis

               -  Conditions that affect lymphocyte counts, such as human immunodeficiency virus
                  (HIV) infection or immunosuppressive therapy

          -  Use of prohibited drugs